Fundamentals of Mathematical Analysis

PROLOGUE: What is Analysis?       Introduction       Precakulus developments in Ancient Greece       Precalculus developments in Western Europe       The discovery of calculus       From calculus to analysis       Conclusion    CHAPTER ONE  Preliminaries       1.1 Logic       1.2 Sets            1.2.1 Laws of the algebra of sets       1.3 Functions    CHAPTER TWO  The Real Numbers       2.1 Numbers            2.1.1 Theorem            2.1.2 Theorem            2.1.3 Theorem            2.1.4 Theorem       2.2 Axioms for the real numbers            2.2.1 The axioms of arithmetic            2.2.2 The axioms of order            2.2.3 The completeness axiom       2.3 The completeness axiom            2.3.1 The Archimedean postulate            2.3.2 Density of the rationals            2.3.3 The principle of mathematical induction    CHAPTER THREE  Sequences       3.1 Convergent sequences            3.1.1 Definition            3.1.2 Rules            3.1.3 Sandwich rule            3.1.4 Composite rule       3.2 Null sequences            3.2.1 Power rule            3.2.2 Basic null sequences       3.3 Divergent sequences            3.3.1 Result            3.3.2 Definition            3.3.3 Rules            3.3.4 Reciprocal rule            3.3.5 Definition            3.3.6 Result       3.4 Monotone sequences            3.4.1 Principle of monotone sequences            3.4.2 Bolzano-Weierstrass theorem            3.4.3 Definition            3.4.4 Theorem    CHAPTER FOUR  Series       4.1 Infinite series            4.1.1 Result            4.1.2 The vanishing condition            4.1.3 Sum rule            4.1.4 Scalar product rule            4.1.5 Result       4.2 Series tests            4.2.1 First comparison test            4.2.2 Second comparison test            4.2.3 D’ Alembert’s ratio test            4.2.4 Altenating series test            4.2.5 Integral test            4.2.6 ’fheore111            4.2.7 l’heorern            4.2.8 The rearrangement rule            4.2. 9 Cauchy product of series       4.3 Power series            4.3.1 Theorem            4.3.2 Arithmetic of power series            4.3.3 Definition            4.3.4 Definition            4.3.5 Definition    CHAPTER FIVE  Continuous Functions       5.1 Limits            5.1.1 Definition            5.1.2 Definition            5.1.3 Rules            5.1.4 Sandwich rule            5.1..5 Composite rule            5.1.6 Theorem       5.2 Continuity            5.2.1 Definition            5.2.2 Definition            5.2.3 Theorem            5.2.4 Rules            5.2.5 Sandwich rule            5.2.6 Composite rule            5.2. 7 Inverse rule       5.3 Theorems            5.3.1 The boundedness property            5.3.2 The intermediate value property            5.3.3 The interval theorem            5.3.4 A fixed point theorem    CHAPTER SIX  Differentiation       6.1 Differentiable functions            6.1.1 Definition            6.1.2 Theorem            6.1.3 Rules            6.1.4 Quotient rule            6.1.5 Sandwich rule            6.1.6 Composite rule            6.1.7 Inverse rule            6.1.8 Definition            6.1.9 Local extremum theorem       6.2 Theorems            6.2.1 Rolle’s theorem            6.2.2 Mean value theorem            6.2.3 The increasing-decreasing theorem            6.2.4 Cauchy’s mean value theorem            6.2.5 L’Hopital’s rule (version A)            6.2.6 Leibniz’s formula            6.2.7 L’Hopital’s rule (version B)       6.3 Taylor polynomials            6.3.1 Definition            6.3.2 The first remainder theorem            6.3.3 Standard series            6.3.4 Definition            6.3.5 Taylor’s theorem            6.3.6 Result            6.3. 7 Classification theorem for local extrema       6.4 Alternative forms of Taylor’s theorem            6.4.1 Taylor’s theorem    CHAPTER SEVEN  Integration       7.1 The Riemann integral            7.1.1 Definition            7.1.2 Definition            7.1.3 Definition            7.1.4 Theorem            7.1.5 Riemann’s condition            7.1.6 Theorem            7.1.7 Definition            7.1.8 Uniform continuity theorem            7.1.9 Theorem            7.1.10 Properties of the Riemann integral            7.1.11 The fundamental theorem of calculus            7.1.12 The integral mean value theorem            7. l .13 Corollary       7.2 Techniques            7.2.1 Integration by parts            7.2.2 Change of variables       7.3 Improper integrals            7.3.1 Definition            7.3.2 Definition            7.3.3 Definition            7.3.4 Comparison test for integrals            7.3.5 Definition    APPENDIX  The Elementary Functions       A.1 Theorem       A.2 The calculus of power series       A.3 Definition       A.4 Definition       A.5 Definition       A.6 Definition       A.7 Definition       A.8 Definition       A. 9 Definition       A.10 Definition       A.11 Definition    Solutions to Exercises    Answers to problems    Index of Symbols    Index